Practicing Success
The differential equation of all circles passing through the origin and having their centres on the x-axis, is |
$y^2=x^2+2 x y \frac{d y}{d x}$ $y^2=x^2-2 x y \frac{d y}{d x}$ $x^2=y^2+x y \frac{d y}{d x}$ $x^2=y^2+3 x y \frac{d y}{d x}$ |
$y^2=x^2+2 x y \frac{d y}{d x}$ |
The equation of the family of circles passing through the origin and having their centres on x-axis is $(x-a)^2+(y-0)^2=a^2$ or, $x^2+y^2-2 a x=0$ Differentiating w.r. to x, we get $2 x+2 y \frac{d y}{d x}-2 a=0 \Rightarrow a=x+y \frac{d y}{d x}$ Substituting the value of a in (i), we get $x^2+y^2-2 x^2-2 x y \frac{d y}{d x}=0$ or, $y^2=x^2+2 x y \frac{d y}{d x}$ as the required differential equation. |